On the phase-space distribution of Bloch eigenmodes for periodic point scatterers

Abstract

Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (to appear); [e-print arXiv:1409.6878 (2014)] who show momentum localisation for zero quasi-momentum in 2-dimensions and is the first result in this direction in 3-dimensions.